How To Science [Part 4: Science]

Observations reveal that when string length doubles, frequency is roughly cut in half—when length is multiplied by factor of 2, frequency is multiplied by factor of 1/2, and vice versa.

Proportionality statements (“f ∝ 1/L”) become useful equations by introducing proportionality constant K, transforming the relationship into f = K/L where K can be calculated from observations.

Using the formula f = K/L with calculated constant K = 6960, specific frequency predictions emerge: 87 Hz at 80cm, 139.2 Hz at 50cm, and 348 Hz at 20cm.

Measurements yield 91 Hz (80cm), 141 Hz (50cm), and 343 Hz (20cm), compared to predictions of 87 Hz, 139.2 Hz, and 348 Hz—producing percent errors of 4.6%, 1.3%, and 1.4%, confirming the hypothesis.

The experimentally validated relationship—frequency of a vibrating string is inversely proportional to its length—is known today as Mersenne’s First Law.

The video synthesizes “the scientific method”—a systematic approach allowing discovery of relationships between mathematics and physical reality through observation, guessing, and experimentation.

The video sets up discovering Mersenne’s Second Law by fixing string length at 60cm and measuring frequency as tension varies, collecting more observations since this relationship is “trickier” than length-frequency.

To compare different guitar string varieties fairly, the video introduces mass per unit length—dividing total mass by full length to get mass per centimeter, allowing comparison independent of string length.

The video challenges discovering Mersenne’s Third Law by measuring frequency for six different guitar string varieties with varying mass per unit length at fixed length and tension.